24 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA Clearly, if / is lower semicontinuous at XG , then — / is upper semicon- tinuous at this point. Hence we need consider only lower semicontinuous functions. If cp is a mapping of a topological space F into E which is continuous at a point y0 E F, and if /is lower semicontinuous at the point ;c0 = (p(y0\ then /Q (p is lower semicontinuous at j0 . In particular, if F is a subspace of E, and if /is lower semicontinuous at a point j0 e F, then so is the function /| F. Examples (12.7.1) A mapping /: E -» R is said to have a relative minimum at a point XQ e E if there exists a neighborhood V of XQ such that/(x) ^/(;c0) for all x E V. If so, then/is lower semicontinuous at the point x0 . This is clearly the case whenever f(x0) = — oo. For each x e R, put f(x) = 0 if x is irrational, and f(x) = \\q if x is rational and equal to p/q (where p, q are coprime integers and q > 0). For each integer n ^ 0, the subspace of rational numbers p/q with q g n is closed in R and discrete. Hence for each irrational x there exists a neighborhood V of x such that/(j;) :g l/n for all ye V, and therefore /is continuous at the point x. On the other hand, /has a relative maximum at each rational point, and is therefore upper semicontinuous on R. (12.7.2) A mapping f: E -» K w /ower semicontinuous on E z/ flfld ow/y z/, for each a e R, ?/ze ^eif /""1(]a, +00]) of points x at which f(x) > a w 0/?e« 1/2 E (or, equivalently, the set/""1^— oo, a]) of points x such that/(jc) ^ a is closed inE). For to say that / is lower semicontinuous on E signifies that, for each a e R, the set/"1Qa, +00]) is a neighborhood of each of its points (3.6.4). If A is any subset of a set E, the characteristic function of A (usually denoted by (pA) is the mapping of E into R such that (pA(x) = 1 for all x e A and cpA(x) = 0 for all x e E — A. So we have (pE = 1, q>0 = 0, and the for- mulas <PA + <PB = 9 A u B + <?A n B > 9 A. n CB = <?A - 9 A 93 > (12.7.3) I<PA - <?B! = <PA o CB + %-n CA » A A A where A, B are any two subsets of E, and (AA) is any family of subsets of E.